In part I a well-known pair of cross phenomena, thermo-osmosis and mechano-caloric effect, in a dense gas is treated by a “thermo-hydrodynamical” method: the FOURIER and NAVIER-STOKES equations are adopted for heat conduction and viscous motion whereas in the boundary conditions MAXWELL’S thermal slip (3.1) and a mechanical surface heat flow (3.7) appear. Both of them are connected by an ONSAGER relation (3.9). So far, the underlying geometry was a special one (circular cylindrical capillary). In part II the boundary conditions problem at the interface between two immiscible fluids is considered in the general case, by the method of continuum non-equilibrium thermodynamics. The fluids are in viscous motion and conducting heat, such that total mass, momentum, angular momentum and energy are conserved. It is assumed that none of these quantities is concentrated in the interface (vanishing densities per unit area). This leads to the global conditions (6.3) for the forces and (6.8) for the normal energy flows at the interface. After the general expression (7.8) for interfacial entropy production is at hand, the global conditions are replaced by stronger local ones in the vein of thermo-hydrodynamics. Two possibilities are considered in this context: the interface a) does not and b) does carry two-dimensional flows of momentum and energy. In case a) the ensuing local boundary conditions are merely mechanical slip for the velocity fields and temperature jump for the temperature fields. In addition to these, in case b) a pair of cross effects naturally comes out, namely thermal slip and mechanical surface heat flow, and a thermal surface heat flow as well. This reassures the results of part I which had been obtained in a somewhat indirect way.